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- Thread starter nycmathguy
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Mark44

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Plot some points. What are f(0), f(1/4), f(1/2), f(2), f(2.5), etc.?Homework Statement::Investigate each limit.

Relevant Equations::See attachment for function.

Investigate each limit.

See attachment.

1. lim f(x) x→2

2. lim f(x) x→1/2

I don't understand this piecewise function.

Also, you've posted a few threads just now with little or no work shown. That's a violation of forum rules. You have to show

- #3

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1) What is f(n)

2) What is f(n+1)

3) What is f(x) for every ##x\in(n,n+1)## for example for x=(2n+1)/2 the midpoint of n and n+1.

- #4

nycmathguy

Sorry but I don't get it. Still lost.

1) What is f(n)

2) What is f(n+1)

3) What is f(x) for every ##x\in(n,n+1)## for example for x=(2n+1)/2 the midpoint of n and n+1.

- #5

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what is f(1) and f(2) equal to for example? Hint: 1 and 2 are integers

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- #7

nycmathguy

I say for (1), the answer is 0.

The answer for (2) is 1.

Yes?

- #8

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Νο, ##f(n)=f(n+1)=1## for all integers n. The function definition tells us that f(x)=1 if x is integer.I say for (1), the answer is 0.

The answer for (2) is 1.

Yes

- #9

nycmathguy

For (1), x tends to an integer. Thus, then f(x) = 1.Νο, ##f(n)=f(n+1)=1## for all integers n. The function definition tells us that f(x)=1 if x is integer.

For (2), x tends to a rational number. Thus, f(x) = 0.

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- #11

nycmathguy

What about the following two cases using the same attachment?Νο, ##f(n)=f(n+1)=1## for all integers n. The function definition tells us that f(x)=1 if x is integer.

Investigate each limit.

1. lim f(x) x→3

2. lim f(x) x→0

For (1), x tends to an integer. Thus, f(x) = 1.

For (2), x tends to 0, which is not an integer.

Thus, f(x) = 1.

Yes?

- #12

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f(1)=f(0)=1

f(x)=0 for all x inbetween 0 and 1.

Then what do you think is the ##\lim_{x\to 0} f(x)## (or ##\lim_{x\to 1} f(x)##..

- #13

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For both cases the limit is 0. (0 is an integer btw).What about the following two cases using the same attachment?

Investigate each limit.

1. lim f(x) x→3

2. lim f(x) x→0

For (1), x tends to an integer. Thus, f(x) = 1.

For (2), x tends to 0, which is not an integer.

Thus, f(x) = 1.

Yes?

- #14

nycmathguy

Homework Statement::Investigate each limit.

Relevant Equations::See attachment for function.

Investigate each limit.

See attachment.

1. lim f(x) x→2

2. lim f(x) x→1/2

I don't understand this piecewise function.

Can you elaborate a little more?

f(1)=f(0)=1

f(x)=0 for all x inbetween 0 and 1.

Then what do you think is the ##\lim_{x\to 0} f(x)## (or ##\lim_{x\to 1} f(x)##..

It's just not sinking in. In fact, Sullivan stated in his book that this is considered a challenging problem.

- #15

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hm ok let me see

If I tell you that f(x)=0 for all x then what is the ##\lim_{x\to 0} f(x)##.

If I tell you that f(x)=0 for all x then what is the ##\lim_{x\to 0} f(x)##.

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- #17

nycmathguy

In that case, it is 0.hm ok let me see

If I tell you that f(x)=0 for all x then what is the ##\lim_{x\to 0} f(x)##.

- #18

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Correct now lets say I tweak the function and the function f is now f(x)=0 for all x EXCEPT for x=0 which I define to be f(0)=1. Do you think that the above limit changes or remains the same?In that case, it is 0.

- #19

nycmathguy

You said except for x = 0. I say the limit is 1?Correct now lets say I tweak the function and the function f is now f(x)=0 for all x EXCEPT for x=0 which I define to be f(0)=1. Do you think that the above limit changes or remains the same?

- #20

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Nope it isn't 1. What is f(x) equal to ,as x tends to 0, for example what is f(0.5), f(0.4), f(0.3) , f(0.2) and so on..You said except for x = 0. I say the limit is 1?

- #21

nycmathguy

So, f(every decimal number you listed) = 0 because decimal numbers are rational and rational numbers are not integers.Nope it isn't 1. What is f(x) equal to ,as x tends to 0, for example what is f(0.5), f(0.4), f(0.3) , f(0.2) and so on..

- #22

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That's correct. So what conclusion can you make from this? where does f(x) tend to as x tends to 0?So, f(every decimal number you listed) = 0 because decimal numbers are rational and rational numbers are not integers.

- #23

nycmathguy

So, f(x) tends to 0 as x-->0.That's correct. So what conclusion can you make from this? where does f(x) tend to as x tends to 0?

- #24

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yes and this is true regardless of what value we choose to give to f(0). As long as f(x)=0 for all ##x\neq 0## .So, f(x) tends to 0 as x-->0.

- #25

nycmathguy

Trust me, I plan to journey through calculus l,ll, and lll. We will see limit questions up the wall.yes and this is true regardless of what value we choose to give to f(0). As long as f(x)=0 for all ##x\neq 0## .

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